### 3.1 Basic principles

Constructing CME models is extremely important, not only because CMEs are a spectacular astronomical phenomenon, but also because they are the main driver for the space weather disturbances that strongly affect our high-tech life. It is emphasized here that any successful model should be based on the combination of observations and magnetohydrodynamic (MHD) theory. The same as any other eruptive phenomenon, CMEs, along with solar flares, involve the energy conversion from one kind to the kinetic, potential, thermal, and nonthermal energies, as well as the radiative energy in flaring loops. It should be noted that there are secondary conversions between different energies, e.g., part of the nonthermal energy would be converted into the thermal energy, which would finally radiate out. They should not be double counted when estimating the CME and flare energies. With the assumption that a typical CME involves a volume of 1024 m3, the energy density of a CME ranges from 10–2 – 10 J m–3. The typical energy density of possible energy sources is shown in Table 1 (Forbes, 2000). We can see that for energetic CME events, which are the most interesting in the space weather context, the only possible source is the magnetic energy, whereas for very weak CME events, thermal and potential energies in the pre-eruption corona may contribute to the CME explosions. In the case that these two sources are available, thermal energy is converted to the CME energy by the work of pressure gradient, similar to the acceleration of solar wind, and the potential energy is converted to the CME energy in the form of buoyancy.

Table 1: Estimates of the coronal energy sources (adapted from Forbes, 2000).
 Form of energy Energy density (J m–3) Observed averaged value Kinetic () 8 × 10–4 n = 1015 m–3, V = 1 km s–1 Thermal () 1 × 10–2 T = 106 K Potential () 5 × 10–2 h = 105 km Magnetic () 40 B = 10–2 T

In those eruptive cases, the CMEs energy comes from the partial release of the magnetic free energy, i.e., the excess energy compared to the potential field with the same flux distribution at the solar photosphere. It is demonstrated that in the case of force-free field that is often applicable in the low corona (Gary, 2001), the magnetic free energy is of the order of the magnetic energy of the corresponding potential field (Aly, 1984). For example, Aly (1991) showed that for a simply connected field the total magnetic energy is less than twice the potential magnetic energy. In the cases when gravity is important, e.g., when a filament is present, or that the gas pressure is not negligible, the total magnetic energy can be derived from the virial theorem (Low, 1999):

which would lead to a higher magnetic free energy. It should be noted here that due to the strong line-tying effect of the solar surface, the photospheric vector magnetogram does not transit to a potential state after the eruption. It remains to be close to a nonlinear force-free field (e.g., Jing et al., 2009; Thalmann and Wiegelmann, 2008), with a certain amount of magnetic free energy and magnetic helicity. Since the whole process is constrained by the conservation of magnetic energy and helicity (Berger, 1984; Aly, 1991), only a part of the free energy, as well as helicity, can be really released from the corona. As a consequence, SXR sigmoids, which are generally believed to be the nonpotential pre-CME structure, may form after the onset of some CMEs (Glover et al., 2001).

Except some slow CMEs which might be accelerated by the ambient solar wind (see Section 4.2), it is well established that many CMEs are due to the rapid release of magnetic energy in the corona. However, the coronal magnetic field can not yet be measured directly. The quantitative study of the coronal magnetic field generally relies on the magnetic extrapolation based on the photospheric magnetograms. However, for a given photospheric magnetogram, the extrapolation of the coronal field is not unique, depending on the methods (DeRosa et al., 2009). More importantly, the extrapolation technique may have smoothed some singular magnetic topologies which are important for the eruptions. Besides, the magnetic field in the photosphere is probably not in a force-free state. Therefore, the understanding of CMEs relies mainly on the multiwavelength imaging and spectroscopic observations. With the following reasons, the physical processes in CMEs are far from been fully resolved: (1) white-light coronagraphs, which observe the CMEs directly, always occult the solar disk and the innermost corona where CMEs originate; (2) coronagraph observations favor the CMEs propagating near the plane of the sky, whereas the CMEs source region can be better diagnosed near the solar disk center; (3) the SXR and EUV emissions of the corona are optically thin, and it is still difficult for the spectrometers to possess both a wide field of view and a high time cadence simultaneously (see Harrison and Lyons, 2000). Despite these difficulties, much progress has been made in the past decades, along with many controversies.